2-torus manifolds, cobordism and small covers

Abstract

Let ${\mathfrak{M}}_n$ be the set of equivariant unoriented cobordism classes of all $n$-dimensional 2-torus manifolds, where an $n$-dimensional 2-torus manifold $M$ is a smooth closed manifold of dimension $n$ with effective smooth action of a rank $n$ 2-torus group $({\mathbb{Z}}_2{)}^n$. Then ${\mathfrak{M}}_n$ forms an abelian group with respect to disjoint union. This paper determines the group structure of ${\mathfrak{M}}_n$ and shows that each class of ${\mathfrak{M}}_n$ contains a small cover as its representative in the case $n=3$.

19 pages, 4 figures